Finiteness theorems for generalized Jacobians with nontrivial torsion

• Published in: Sbornik. Mathematics Vol. 216; no. 4; pp. 560 - 577
• Main Authors: Platonov, Vladimir Petrovich, Zhgoon, Vladimir Sergeevich, Fedorov, Gleb Vladimirovich
• Format: Journal Article
• Language: English
• Published: 2025
• ISSN: 1064-5616; 1468-4802
• DOI: 10.4213/sm10142e

Consider a curve $\mathcal C$ defined over an algebraic number field $k$. This work is concerned with the number of generalized Jacobians $J_{\mathfrak{m}}$ of $\mathcal C$ associated with moduli $\mathfrak{m}$ defined over $k$ such that a fixed class of finite order in the Jacobian $J$ of $\mathcal C$ is lifted to a torsion class in the generalized Jacobian $J_{\mathfrak{m}}$. On the one hand it is shown that there are infinitely many generalized Jacobians with the above property, and on the other hand, under some additional constraints on the support of $\mathfrak{m}$ or the structure of $J_{\mathfrak{m}}$, it is shown that the set of generalized Jacobians of this type is finite. In addition, it is proved that there are finitely many generalized Jacobians which have a lift of two given divisors to classes of finite orders in $J_{\mathfrak{m}}$. These results are applied to the problem of the periodicity of continued fractions in the field of formal power series $k((1/x))$ constructed for special elements of the function field $k(\widetilde{\mathcal{C}})$ of a hyperelliptic curve $\widetilde{\mathcal{C}}\colon y^2=f(x)$. In particular, it is shown that for each $n \in \mathbb N$ there is a finite number of monic polynomials $\omega(x) \in k[x]$ of degree at most $n$ such that the element $\omega(x) \sqrt{f(x)}$ has a periodic expansion in a continued fraction. Bibliography: 14 titles.